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Chapter 27: Problem 25

A diffraction pattern forms when light passes through a single slit. The wavelength of the light is 675 nm. Determine the angle that locates the first dark fringe when the width of the slit is (a) \(1.8 \times 10^{-4} \mathrm{m}\) and \(\quad(\mathbf{b}) 1.8 \times 10^{-6} \mathrm{m}\)

Short Answer

Expert verified
For (a) \( \theta \approx 0.215 \text{ degrees} \); for (b) \( \theta \approx 22.02 \text{ degrees} \).

Step by step solution

01

Understand the Problem

The problem asks to find the angle that corresponds to the first dark fringe in a diffraction pattern. This occurs when light passes through a single slit and is related to the slit width and light wavelength.
02

Apply the Formula for Single-Slit Diffraction

The formula for locating dark fringes in single-slit diffraction is given by \( a \sin \theta = m \lambda \), where \( a \) is the slit width, \( \theta \) is the angle of the dark fringe, \( m \) is the order of the dark fringe (1 for the first dark fringe), and \( \lambda \) is the wavelength of light.
03

Calculate the Angle for Part (a)

For part (a), where the slit width \( a = 1.8 \times 10^{-4} \: \mathrm{m} \) and \( \lambda = 675 \: \mathrm{nm} = 675 \times 10^{-9} \: \mathrm{m} \), we use the formula \( \sin \theta = \frac{m \lambda}{a} = \frac{1 \times 675 \times 10^{-9}}{1.8 \times 10^{-4}} \). Calculating this gives \( \sin \theta = 0.00375 \). Therefore, \( \theta = \arcsin(0.00375) \approx 0.215 \) degrees.
04

Calculate the Angle for Part (b)

For part (b), where the slit width \( a = 1.8 \times 10^{-6} \: \mathrm{m} \), we again apply the formula \( \sin \theta = \frac{1 \times 675 \times 10^{-9}}{1.8 \times 10^{-6}} \). Calculating this gives \( \sin \theta = 0.375 \). Therefore, \( \theta = \arcsin(0.375) \approx 22.02 \) degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diffraction Pattern
When light shines through a narrow opening like a single slit, it doesn't just pass straight through. Instead, the light spreads out and forms a pattern of alternating dark and bright regions. This phenomenon is known as a diffraction pattern. This pattern demonstrates the wave nature of light. Instead of moving in straight lines, the light waves spread out, interfere, and create areas with no light (dark fringes) and areas with intense light (bright fringes). The spacing and intensity of these fringes provide valuable insights into the properties of the light and the slit itself.
Wavelength of Light
The wavelength of light is a key factor in determining the characteristics of a diffraction pattern. Wavelength ( \( \lambda \) ) is the distance between successive peaks of a light wave. For this exercise, we are considering light with a wavelength of 675 nanometers (nm), which is part of the visible spectrum.
  • Wavelength affects how much the light will spread out after passing through the slit.
  • Longer wavelengths lead to wider spread and thus, wider diffraction patterns.
  • Shorter wavelengths cause a tighter, more compact pattern.
Knowing the wavelength helps you calculate the position of dark and bright fringes in the pattern.
Angle of Dark Fringe
In the diffraction pattern, dark fringes occur at specific angles where the light waves cancel each other out due to destructive interference. The angle of a dark fringe, especially the first one, is what this exercise focuses on. To find this angle, the relationship for single-slit diffraction is used: \[ a \sin \theta = m \lambda \] Where:
  • \( a \) is the slit width,
  • \( \theta \) is the angle of the dark fringe,
  • \( m \) is the order number of the fringe (1 for the first dark fringe),
  • \( \lambda \) is the wavelength of light.
This equation allows us to calculate the precise angle where the first dark fringe will appear, by solving for \( \theta \) using the known values of \( a \) and \( \lambda \).
Slit Width
The slit width ( \( a \) ) plays a crucial role in shaping the diffraction pattern. In the given problem, the widths provided are \(1.8 \times 10^{-4} \mathrm{m}\) and \(1.8 \times 10^{-6} \mathrm{m}\).
  • A larger slit width will cause the diffraction pattern to be narrower, with less spread between the fringes.
  • A smaller slit width increases the divergence of light, leading to a wider spread in the pattern.
  • The size of the slit also directly determines where the dark and bright fringes appear. The equation \( a \sin \theta = m \lambda \) directly relates slit width to the angle of diffraction.
Understanding the impact of slit width helps in predicting how the diffraction pattern will look under different conditions.

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Most popular questions from this chapter

A slit has a width of \(W_{1}=2.3 \times 10^{-6} \mathrm{m} .\) When light with a wavelength of \(\lambda_{1}=510 \mathrm{nm}\) passes through this slit, the width of the central bright fringe on a flat observation screen has a certain value. With the screen kept in the same place, this slit is replaced with a second slit (width \(W_{2} ),\) and a wavelength of \(\lambda_{2}=740 \mathrm{nm}\) is used. The width of the central bright fringe on the screen is observed to be unchanged. Find \(W_{2} .\)

How many dark fringes will be produced on either side of the central maximum if light \((\lambda=651 \mathrm{nm})\) is incident on a single slit that is \(5.47 \times 10^{-6} \mathrm{m}\) wide?

An inkjet printer uses tiny dots of red, green, and blue ink to produce an image. Assume that the dot separation on the printed page is the same for all colors. At normal viewing distances, the eye does not resolve the individual dots, regardless of color, so that the image has a normal look. The wavelengths for red, green, and blue are \(\lambda_{\mathrm{red}}=660 \mathrm{nm}\) \(\lambda_{\text { green }}=550 \mathrm{nm},\) and \(\lambda_{\mathrm{blue}}=470 \mathrm{nm} .\) The diameter of the pupil through which light enters the eye is 2.0 \(\mathrm{mm}\) . For a viewing distance of \(0.40 \mathrm{m},\) what is the maximum allowable dot separation?

A flat observation screen is placed at a distance of 4.5 m from a pair of slits. The separation on the screen between the central bright fringe and the first-order bright fringe is 0.037 m. The light illuminating the slits has a wavelength of 490 nm. Determine the slit separation

Violet light (wavelength \(=410 \mathrm{nm} )\) and red light (wave length \(=660 \mathrm{nm}\) ) lie at opposite ends of the visible spectrum. (a) For each wavelength, find the angle \(\theta\) that locates the first-order maximum produced by a grating with 3300 lines/cm. This grating converts a mixture of all colors between violet and red into a rainbow-like dispersion between the two angles. Repeat the calculation above for \((\mathbf{b})\) the second order maximum and \((\mathbf{c})\) the third-order maximum. (d) From your results, decide whether there is an overlap between any of the "rainbows" and, if so, specify which orders overlap.
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